One computational method which uses hexahedral meshes is Finite Element Analysis (FEA). FEA is the process of creating a finite element mesh (“FEM”), which represents a physical domain upon which some physical phenomenon is to be analyzed. These domains can be broken up into either two dimensional (“2D”) or three dimensional (“3D”) domains. 3D domains represent the full-3D dimensions of an actual 3D domain. 3D domains are most often modeled with either tetrahedral or hexahedral elements. Less often, 3D domains are modeled with pyramid or wedge elements. FIG. 1 illustrates these four basic element types.
2D domains represent a physical phenomenon which is geometrically located in some kind of surface (either planar or non-planar), such as surface wave front propagation in liquids, or a thin sheet metal object such as the hood of a car. In addition, 2D domains are used to represent a simplification of a 3D domain, such as a cross-section of a 3D domain. 2D domains are most often modeled with either quadrilateral or triangular elements. FIG. 2 illustrates these 2D element types.
FEMs are typically composed of a single element type. For example, a hexahedral mesh is composed of only hexahedral elements. A “hybrid” mesh is a mesh composed of more than a single element type. For most FEA solvers, a non-hybrid mesh is preferred. Many FEA solvers do not support hybrid meshes.
FEA and all of its variations are an important part of current design through analysis processes. Thanks to its flexibility, FEA successfully reproduces a large spectrum of physical phenomena. FEA is performed by first decomposing a volumetric model geometry into a set of face-connected elements, usually tetrahedral or hexahedral finite elements. Each element is defined by connecting a set of nodes, which define discrete position vectors throughout the model domain. The collection of finite elements forms a finite element mesh, which provides a discrete approximation to the object being analyzed. This allows the solution of the governing equations to be approximated with simple shape functions, typically linear or quadratic functions, over each individual element. A system of linear equations follows with the unknowns being approximations to the solution at the discrete nodal positions. These nodal values, combined with the element shape functions, provide an approximation of the solution on the system as a whole.
Hexahedral mesh generation is an important bottleneck preventing broader use of FEA, often consuming as much as 70-80% of the overall design through analysis process. Hexahedral mesh generation is particularly difficult when the model being meshed is composed of multiple different volumes which must behave as a single solid. Multi-volume models arise in two different scenarios. First, the model represents a multi-part mechanism, which has some parts bonded together to act as a single object. Second, the model is a single physical component, which has been decomposed geometrically in order to simplify the meshing process. For example, FIG. 3A illustrates a model with a single component. In FIG. 3B, the part is decomposed allowing each partition to be meshed with hexahedral elements as illustrated in FIG. 3C and then joined as illustrated in FIG. 3D.
Regardless of the origins of the multi-part model, modeling of the interfaces between adjacent components is particularly important. Two or more spatially adjacent components often must behave as a single component since their interfaces are welded or glued together. A conforming mesh across such interfaces ensures inter-element continuity in the finite element shape functions resulting in a smooth and accurate interpolation of the numerical solution. However, requiring a conforming mesh severely restricts hexahedral mesh generators. Satisfying the resulting constraints requires a time consuming global linear programming problem of interval assignment, and requires all surfaces to be meshed in a single serial process. In addition, strict hexahedral topology requirements paired with conforming interface constraints, often leads to an over-constrained system and meshing failure. Other known hexahedral finite element methods are completely unable to generate conforming interfaces on multi-volume models.
Optionally, interfaces can be left non-conforming by using artificial constraints such as multi-point constraints, tied contacts, gap elements, or mortar formulations. Although these artificial constraint techniques have matured significantly in recent years, non-conforming meshes break shape function inter-element continuity, resulting in solution quality degradation, discontinuous solution fields and/or adverse effects on solution convergence. In addition, these artificial constraints introduce additional equations to the system increasing the amount of time required to solve the system. Thus, non-conforming interface conditions should be avoided in critical regions of the model. Conforming interfaces are typically preferred, whenever possible.